Hello, karliekay!

Join two 1-by-1 squares to form a rectangle.

. . Next, draw a larger square along a long side of the rectangle.

. . Continue this process.

a) Write the sequence of the lengths of the sides of the squares you added at each step.

. . Begin the sequence with two original squares.

We start with one square.

Code:

*---*
1 | |
*---*
1

Add a square:

Code:

*---*
| |
*---*
|:::| 1
*---*
1

Add a square:

Code:

*---*-------*
1 | |:::::::|
*---*:::::::| 2
1 | |:::::::|
*---*-------*
2

Add a square.

Code:

*---*-------*
1 | | |
*---* | 2
1 | | |
*---*-------*
|:::::::::::|
|:::::::::::|
3 |:::::::::::| 3
|:::::::::::|
*-----------*
3

Add a square.

Code:

*---*-------*-----------------*
1 | | |:::::::::::::::::|
*---* |:::::::::::::::::|
1 | | |:::::::::::::::::|
*---*-------*:::::::::::::::::|
| |:::::::::::::::::| 5
| |:::::::::::::::::|
3 | |:::::::::::::::::|
| |:::::::::::::::::|
*-----------*-----------------*
3 5

[tex]\text{It appears that the sequence is:}\;1,\,1,\,2,\,3,\,5,\,8,\,13\:\hdots[/tex]

b) Write a recursive formula for the sequence of lengths added.

[tex]\text{Each term is the sum of the preceding two terms.}[/tex]

. . [tex]F_n \;=\;F_{n-1} + F_{n-2}[/tex]

c) Identify the sequence in b.

[tex]\text{This is the Fibonacci sequence.}[/tex]

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